Elasto-hydrodynamics of quasicrystals with a crack under sudden impacts

ABSTRACT

In this paper, a mode-III crack in one-dimensional (1D) hexagonal quasicrystals subjected to anti-plane impact loading is analyzed. The elasto-hydrodynamics of quasicrystals is adopted, where the phonon field obeys wave equation, and the phason field obeys diffusion equation. By introducing a new auxiliary function, the coupled wave-diffusion equations are converted to a single higher-order partial differential equation. With the aid of the Laplace transform, an associated mixed initial-boundary value problem is reduced to two sets of dual integral equations, and then transformed into two coupled Fredholm integral equations of the second kind. Numerical results of transient phonon and phason stress intensity factors and crack-centre displacement jump are obtained through the numerical inversion of the Laplace transform and are presented graphically to show the influences of the phason.     

Interaction of a dislocation with an elliptical hole in icosahedral quasicrystals

The interaction between a dislocation and an elliptical hole in icosahedral quasicrystals is considered. An explicit expression for the complex potential is derived using the extended Stroh formalism. Based on the conformal mapping method and a perturbation technique, closed-form solutions are obtained. The field intensity factors at a crack tip and the image forces on the dislocation arising from the crack are calculated. The effects of phonon–phason elastic coupling on the mechanical behavior are also observed.     

X-ray study of phason strain in Al-based quasicrystals

Abstract

X-ray powder diffraction spectra have been measured for melt-spun Al-Si–Mn, Al-Cu–Fe and Al-Li–Cu and Bridgman-grown Al-Li-Cu quasicrystals to investigate the phason strain frozen during the solidification process. For the melt-spun samples, most of the peaks have a shoulder or a tail which is attributed to the anisotropic linear phason strain. The peak shapes for the three melt-spun samples are essentially the same, which indicate that the same type of linear phason is quenched in these samples. In contrast, symmetric peak shapes without shoulders nor tails are seen for the Bridgman-grown Al-Li–Cu, indicating that it contains little linear phason strain.

Our findings are discussed with respect to the three linear phason models which can be derived by degradation of the symmetry from ideal icosahedral group m35 to its maximal subgroups 3m, m3 and 5m. Of these three, the model for 3m reproduces best the observed shapes and widths of the peaks.     

Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from diffuse scattering measurements

The diffuse scattering in the diffraction pattern of the icosahedral Al-Pd-Mn quasicrystalline phase has been measured on an absolute scale by X-ray and neutron scattering on single-grain samples. Most of the diffuse scattering can be interpreted in the framework of the elasticity theory of icosahedral quasicrystals considering only phason fluctuations. At room temperature the absolute values of the K₁/k_BT and K₂/k_B T phason elastic constants are of the order of 0.06 and 0.031atom^-1. The amount of diffuse scattering intensity is insensitive to the sample annealing treatment.     

Comparison of quasicrystalline (T2) and crystalline (R) structures in AlCuLi using high-resolution X-ray diffraction

Abstract

A high-resolution X-ray scattering experiment has been performed on quasicrystalline (T2) and crystalline (R) phases of AlCuLi alloys. Peak widths determined from the diffraction profile of the R phase are found to scale with momentum transfer G, similar to the strain-broadening effect seen in conventional crystals with small unit cells. A narrowing of peak widths on annealing is also observed. On the other hand, diffraction peak widths in I-phase samples obtained by rapid solidification are found to scale with phason momentum G ₁, similar to that observed in large faceted crystals. Thermal annealing of the I phase is found to have no effect on peak widths. Our findings are discussed in the light of models for crystals and quasicrystals.     

Dislocations in quasicrystals and their interaction with cluster-like obstacles

A dislocation moving through a quasicrystal leaves in its wake a fault denoted a phason wall. For a two-dimensional model quasicrystal the disregistry energy of this phason wall is studied to determine possible Burgers vectors of the quasicrystalline structure. Unlike periodic crystals, the disregistry energy is an average quantity with large fluctuations on the atomic scale. Therefore the dislocation core structure and mobility cannot be linked to this quantity, e.g. by a Peierls-Nabarro model. Atomistic simulations show that dislocation motion is controlled by local obstacles inherent to the atomic structure of the quasicrystal.     

A general solution to Mosier's oblique procrustes problem

Browne provided a method for finding a solution to the normal equations derived by Mosier for rotating a factor matrix to a best least squares fit with a specified structure. Cramer showed that Browne's solution is not always valid, and proposed a modified algorithm. Both Browne and Cramer assumed the factor matrix to be of full rank. In this paper a general solution is derived, which takes care of rank deficient factor matrices as well. A new algorithm is offered.     

From the decagonal phase to a related crystalline one

Abstract

We present here a rotation matrix in five-dimensional space which connects the decagonal phase to a tetragonal crystalline related phase through a one-dimensional quasicrystals range by means of a phason-phonon coupling. Phason and phonon fields are determined in a general way and, from this result, a simple linear approximation is given. Continuous evolution density-wave patterns are shown.     

Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer

A large class of statistical decision models for performance in simple information processing tasks can be described by linear, first-order, stochastic differential equations (SDEs), whose solutions are diffusion processes. In such models, the first passage time for the diffusion process through a response criterion determines the time at which an observer makes a decision about the identity of a stimulus. Because the assumptions of many cognitive models lead to SDEs that are time inhomogeneous, classical methods for solving such first passage time problems are usually inapplicable. In contrast, recent integral equation methods often yield solutions to both the one-sided and the two-sided first passage time problems, even in the presence of time inhomogeneity. These methods, which are of particular relevance to the cognitive modeler, are described in detail, together with illustrative applications. Copyright 2000 Academic Press.     

Sources of difficulty in understanding newtonian dynamics

A series of force and motion problems was presented to 40 high school science students. The results revealed that the majority of the students only partially understood the formalisms of Newtonian physics. Many had difficulty determining the effect that a force would have on an object' speed of motion and many neglected to take into account thecurrent speed of motion when attempting to predict how a force would alter an obiect's direction of motion. The students' justifications for their erroneous answers suggest that they utilize diverse, and often inconsistent, ideas when solving force and motion problems. These include (a) faulty beliefs about force and motion which have been derived from living in a world where unseen frictional forces operate; (b) partially understood examples from physics class; and (c) knowledge of the properties of scalar arithmetic, some of which do not transfer to vector arithmetic. The students thus employ components of their knowledge, which seem relevant to the solution of force and motion problems, but which often have properties that conflict with the implications of Newton's laws of motion.     

Determination of atomic structure of phason planes in the ξ′-Al–Pd–Mn phase

The atomic structures of specific types of linear defects (phason lines) and planar defects (phason planes) in the complex metallic alloy phase ξ′-Al–Pd–Mn have been determined by high-resolution electron microscopy (HREM) and theoretical HREM simulation. The results show that a representational atomic structural model for phason planes can be constructed by introducing a shift between two parts of the perfect crystalline structure using a translation vector of r ?=?(1/2) a ?+?(1/2τ) c . This typical phason plane is normally parallel to the (001) plane of the ξ′-Al–Pd–Mn phase and consists of phason lines, which are arranged side-by-side with their linear direction parallel to the [010] axis. HREM simulations, based on the structural model for both edge-on and inclined types of phason lines, agree well with the experimental results. Taking into account the fact that the structural difference between various curved phason planes arises from the variation in the arrangement of individual phason lines, the atomic structures of the edge-on and inclined phason lines can be used to explain the various curved phason planes frequently observed in the ξ′-Al–Pd–Mn phase.     

A note on the derivation of the general latent class model

A general formulation of the latent structure principle is suggested, from which it is possible to derive Lazarsfeld's accounting equations in their most general form. The basic equations of Gibson's latent profile model can thence be derived in a single step.     

On the origins of dynamical awareness

An inquiry into the origins of dynamical awareness is conducted. Particular attention is given to a theory that postulates that impressions of dynamical quantities are derived from and structured by lawful physical relations. It is shown that impressions of dynamical quantities are not generally correlated with the values that these quantities take in the equations of motion but rather are highly correlated with simple ratios of kinematic quantities or with specific kinematic features that do not specify the underlying dynamics. It is argued that kinematic information, to the extent that it is used, is used heuristically, and its availability for dynamical analysis is constrained by general principles of organization. A formal analysis of the physical organization implicit in the specification of dynamical invariants is given and compared with types of perceptual organization that are observed.     

Maximizing predictive efficiency for a fixed total testing time

For any fixed total time of testing it is possible, through proper item-and-time allotment, to combine tests into a battery so that the multiple correlation with a pre-assigned criterion will be maximized. By holding constant the ratio of the length in number of items to the time length for each test, a set of general equations has been derived which will yield this maximum value of the multipleR and will enable one to determine, in any given case, the optimal fraction of total testing time that should be devoted to each type of test under consideration. The set of general equations is applied to a two-test-battery problem to obtain the optimal length of each type of test for one hour total testing time. If two other tests had been selected for the two-test sample problem, different subdivisions of the total time would generally occur. The manner in which the results would change when using other tests with different initial reliability, validity, and intercorrelation values is briefly presented. Some general implications of this method of battery development are also discussed.The writer is indebted to Max Woodbury for his assistance and especially to Dr. N. J. F. Van Steenberg and Dr. Anna S. Henriques, who provided valuable guidance and aid in the development of the solution to this problem. This paper is a revision of a thesis submitted in 1939 at the University of Utah in partial fulfillment of the requirements for the master's degree.     

The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra

The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly. Furthermore, decreased solution time was accompanied by a decrease in the number of mathematical errors. Both of these findings were specific to problems identical in structure to the initial ones. It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practice and exposure to a wider range of schemas.     

An analytic solution of initial boundary value problem for 3D quasicrystals in half space

In this article, initial boundary value problem for 3D quasicrystals in half space is considered. An analytic method is proposed for special form of initial conditions and nonhomogeneous term. It is explained that a weak solution of the problem can be constructed in the similar form of data by using symbolic calculations.     

Fitting the ratcliff diffusion model to experimental data

Many experiments in psychology yield both reaction time and accuracy data. However, no off-the-shelf methods yet exist for the statistical analysis of such data. One particularly successful model has been the diffusion process, but using it is difficult in practice because of numerical, statistical, and software problems. We present a general method for performing diffusion model analyses on experimental data. By implementing design matrices, a wide range of across-condition restrictions can be imposed on model parameters, in a flexible way. It becomes possible to fit models with parameters regressed onto predictors. Moreover, data analytical tools are discussed that can be used to handle various types of outliers and contaminants. We briefly present an easy-touse software tool that helps perform diffusion model analyses.     

Numerical evaluation of the validity of experimental proofs in biology

Summary This paper suggests a method to calculate a degree of validity for the proof of a statement which is derived from empirical statements by means of logic conclusions. The empirical statements are assumed not to be completely valid or their validity to be doubtful. The suggested rules are consistent with two-valued logic, yield decreasing validities with increasing number of applications of modus ponens and obey the law of the excluded middle. The actual calculation of validity values, the relation of the suggested method to some truth tables of multi-valued logic and to fuzzy logic are discussed.     

Visually guided collision avoidance and collision achievement

To survive on today's highways, a driver must have highly developed skills in visually guided collision avoidance. To play such games as cricket, tennis or baseball demands accurate, precise and reliable collision achievement. This review discusses evidence that some of these tasks are performed by predicting where an object will be at some sharply defined instant, several hundred milliseconds in the future, while other tasks are performed by utilizing the fact that some of our motor actions change what we see in ways that obey lawful relationships, and can therefore be learned. Several monocular and binocular visual correlates of the direction of an object's motion relative to the observer's head have been derived theoretically, along with visual correlates of the time to collision with an approaching object. Although laboratory psychophysics can identify putative neural mechanisms by showing which of the known correlates are processed by the human visual system independently of other visual information, it is only field research on, for example, driving, aviation and sport that can show which visual cues are actually used in these activities. This article reviews this research and describes a general psychophysically based rational approach to the design of such field studies.     

A Faradayan Principle for Selecting Classical Field Theories

Faraday’s field concept presupposes that field stresses should share the axial symmetry of the lines of force. In the present article, the field dynamics is similarly required to depend only on field properties that can be tested through the motion of test‐particles. Precise expressions of this ‘Faradayan’ principle in field‐theoretical language are shown to severely restrict the form of classical field theories. In particular, static forces must obey the inverse square law in a linear approximation. Within a Minkowskian and Lagrangian framework, the Faradayan principle automatically leads to Maxwell’s theory of electromagnetism and to Einstein’s theory of gravitation, without appeal to the equivalence principle. A comparison is drawn between this, Feynman’s, and Einstein’s way to arrive at general relativity.